A335384 Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.

6, 48, 168, 180, 480, 2016, 3528, 5760, 11232, 13200, 20160, 26208, 61200, 78336, 123120, 181440, 267168, 374400, 511056, 682080, 892800, 1014816, 1488000, 1822176, 2755200, 3337488, 4773696, 5644800, 7738848, 9999360, 11908560, 13615200, 16511040, 19845936, 24261120, 25048800, 28003968

Offset

1,1

Comments

GL(m,q) is the general linear group, the group of invertible m X m matrices over the finite field F_q with q = p^k elements.

By definition, all fields must contain at least two distinct elements, so q >= 2. As GL(1,q) is isomorphic to F_q*, the multiplicative group [whose order is p^k-1 (A181062)] of finite field F_q, data begins with m >= 2.

Some isomorphisms (let "==" denote "isomorphic to"):

a(1) = 6 for GL(2,2) == PSL(2,2) == S_3.

a(2) = 48 for GL(2,3) that has 55 subgroups.

a(3) = 168 for GL(3,2) == PSL(2,7) [A031963].

a(11) = 20160 for GL(4,2) == PSL(4,2) == Alt(8).

Array for order of GL(m,q) begins:

=============================================================

m\q | 2 3 4=2^2 5 7

-------------------------------------------------------------

2 | 6 48 180 480 2016

3 | 168 11232 181440 1488000 33784128

4 | 20160 24261120 2961100800 116064000000 #GL(4,7)

5 |9999360 #GL(5,3) ... ... ...

References

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.

Daniel Perrin, Cours d'Algèbre, Maths Agreg, Ellipses, 1996, pages 95-115.

Links

Table of n, a(n) for n=1..37.

Wikipedia, General linear group

Formula

#GL(m,q) = Product_{k=0..m-1}(q^m-q^k).

Example

a(1) = #GL(2,2) = (2^2-1)*(2^2-2) = 3*2 = 6 and the 6 elements of GL(2,2) that is isomorphic to S_3 are the 6 following 2 X 2 invertible matrices with entries in F_2:
  (1 0)   (1 1)   (1 0)   (0 1)   (0 1)   (1 1)
  (0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
a(2) = #GL(2,3) = (3^2-1)*(3^2-3) = 8*6 = 48.
a(3) = #GL(3,2) = (2^3-1)*(2^3-2)*(2^3-2^2) = 168.

Crossrefs

Cf. A059238 [GL(2,q)].

Cf. A002884 [GL(m,2)], A053290 [GL(m,3)], A053291 [GL(m,4)], A053292 [GL(m,5)], A053293 [GL(m,7)], A052496 [GL(m,8)], A052497 [GL(m,9)], A052498 [GL(m,11)].

Cf. A316622 [GL(n,Z_k)].

Sequence in context: A250274 A167547 A244726 * A331668 A005353 A047927

Adjacent sequences: A335381 A335382 A335383 * A335385 A335386 A335387

Keyword

nonn

Author

Bernard Schott, Jun 04 2020

Status

approved